Everyone who cares about mathematics should watch this hilarious and educational video:
If you don’t know who Serre is, read a bit about him before watching the video. You’ll enjoy it more.
I found out about this on Alain Connes’ blog.
Allen Knutson will be amused to hear what Serre has to say about ‘principle bundles’.
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Hmmm. Doesn’t look like it’s in quite the same genre as Science Made Stupid, but perhaps they share a similar spirit.
Blake mentioned:
Science Made Stupid
I never did read that book, although I did read the sequel, Cvltvre Made Stvpid. It was pretty funny!
I see that he has a new(er) book, Minim. These are supposed to be pithy but useless sayings (in contrast to maxims), but some of them are deeper than they may appear at first glance! For example, wise mathematicians (like John) have often used the minim on page 11 as an example of noncommutativity.
I remember Science Made Stupid. Max Reade gave it to me way back when. I wonder if it’s still around my parents’ basement somewhere…
Okay, just watched it. Great fun! Principle bundles have moral fibres. They’re the good, good, good, good fibrations! I’ll never forget that one.
On a more serious note, I was quite disturbed by some of the things Serre said regarding proofs in homological algebra and “identifications” like which are strictly speaking not identifications, but injective homomorphisms.
I’ve been a bit depressed lately about this stuff. For one thing, I’ve been reading Bott and Tu. You’ll see all sorts of equality signs there where they really mean isomorphism. Sometimes its not even a canonical isomorphism.
Take for instance, Proposition 23.9 about how every bundle “is” a pullback of one over a Grassmannian:
Let be a rank complex vector bundle over a differentiable manifold …. then there is a map from to some Grassmannian such that is the pullback under of the universal quotient bundle , that is, .
There’s more… at the end of the proof we find:
We can identify with , and with .
The dodgy things here are the statements “”, the use of the words “this map ” after the proposition (in violation of Serre’s advice), and the cavalier identification of with .
The depressing thing is that: it’s all completely clear what he’s saying. If we included the gory details about isomorphisms satisfying commutative diagrams of their own, just imagine how long and horrible Bott and Tu’s book would become!
You’ll find algebraic topologists doing this all the time. “ is the colimit of such and such”, or “the classifying space”, and so on.
I know that as -cafe regulars, we have been taught by John and others that “the” is always understood to mean “up to canonical isomorphism satisfying some natural commuting diagram”.
But somehow I still find it very awkward…
This business of “harmless identifications” has been exposed by category theory to be not-so-harmless. Who would have thought that the innocent isomorphism of tensor products of vector spaces,
would have been non-trivial? If you think its not: it is, because its precisely this associator which distinguishes the representation categories of the dihedral and quaternionic groups and . The other way to proceed is by anafunctors. But somehow these also give me the heeby jeebies, because its kind of a non-algebraic description. Tensor product is no longer an operation in the strictest sense of the word.
To give a more concrete example: I work with 2-representations of groups on 2-Hilbert spaces. One of the things you need to do right at the beginning is to choose , for each morphism of 2-Hilbert spaces, a morphism which is left and right adjoint to - and yes, you must choose the unit and co-unit maps too!
That gives me the heeby-jeebies (axioms of choice nightmares), but it seems the only way to proceed… at least, in an “algebraic” fashion.
As the ladder of higher categories climbs ever higher and higher into the sky, I sense that mathematical terminology is less and less able to cope elegantly with the notion of “equality”. Sometimes it makes me want to quit this game, and go and do some ordinary maths, where there’s not a sinister presence in the corner which cracks its whip everytime you assert the “equality” of two things when you should have said “isomorphism satisfying some diagram” or worse… you know what I mean.
Think about the word “canonical”. What on earth does it mean nowadays? Suppose there exists a left adjoint to a functor . Most mathematicians will tell you that is given “canonically”. When pushed, they’ll admit it only exists up to a unique natural isomorphism which satisfies some diagram. See? In the good old days, the word “canonical” used to be a very strong word, which meant something like “this map really exists, independent of any choices”. Nowadays we have to use the term “supercanonical” for such a concept. Imagine how awkward it will be when we’re up to 5-categories!
I don’t see why you’re depressed.
We’re trying to understand the universe, which involves understanding the idea of ‘being’, or ‘existence’. I don’t see how we can understand existence until we understand what ‘is’ means. But this turns out to be subtler than you might expect!
In fact, it takes hard work to deeply understand words like ‘equals’, ‘identify’ and even ‘the’, which many mathematicians naively fling around without careful thought.
The example of versus is indeed a great one. You can only recover these groups from their representations if you know precisely what you mean by ‘is the same as’ when you say the representation ‘is the same as’ . All the information is hiding in the isomorphism: the associator!
Even more exciting, to me, is how 3d Riemannian quantum gravity is built from the associator in . The passage of time changes things… but only to something isomorphic. That’s a lesson we can learn from Heraclitus. But in 3d quantum gravity, it becomes very concrete: the passage of time comes from an associator!
That’s a truly staggering idea. That’s is why I got interested in -categories.
I fully expect that a deeper understanding of the universe will require a deeper refinement of basic concepts like ‘equals’, ‘the’ and so on… lots more.
After all, the universe is a truly remarkable and mysterious thing: it is! What’s it really like? What sort of thing can exist, all on its own? Just juggling equations in a technically adept way is unlikely to get us very far in understanding big questions like these.
You almost sound like you wish math could be ‘just business as usual’ — “just ordinary math” as you put it, without deep conceptual problems standing in the way of progress. I feel the opposite, of course. To me, that would make math pretty dull!
But, I don’t think we should spend all our time worrying about conceptual issues. The best diet is a mix of conceptual issues and concrete calculations. I’m spending tons of time these days studying Hecke algebras and cohomology rings of flag manifolds with Todd Trimble and Jim Dolan. There’s a lot of higher category theory in this stuff… but also lots of pretty pictures and hard numbers. Maybe you’re suffering from a lack of calculations in your daily regimen? Like physical exercise, calculations are a great mood booster.
Sometimes it makes me want to quit this game, and go and do some ordinary maths, where there’s not a sinister presence in the corner which cracks its whip everytime you assert the “equality†of two things…
I hope you’re not referring to me! 
Seriously: I don’t think things are going to be a big complicated mess when we sort them out. We just need to expect that our future ways of talking will be quite different than the way we talk now. Have you ever read, oh, say, Diophantus? His stuff is really hard to read. By the time we figure out this -category business properly, our ways of thinking and talking may make our present-day ways look as old-fashioned as Diophantus does now.
(Some people act like the history of thought is almost over: just a few details left here and there that need polishing. I think it’s just starting… with any luck.)
You’ve said so before, but this time it finally sank in. Wow!
“in 3d quantum gravity, it becomes very concrete: the passage of time comes from an associator!”
I am an anti-expert of writing Math badly. My ratio of journal publications to online Math and Physics publications + conference proceedings publications means that I have a LOT to learn from you and your colleagues.
I have had reason to read Diophantus and Fibonacci (translated into English but with actual symbols as used, sexigesimal in the case of Fibonacci’s extremely accurate numerical solution to a cubic, by a methodology that we have not been able to reconstruct). And how did those Islamic architects manage to beat Roger Penrose to quasicrystals and pentagonal symmetry by 500 years? Surely they did not know that these quasicrystals are 3-D projections of regular crystals in dimension 4?
John B. wrote:
I fully expect that a deeper understanding of the universe will require a deeper refinement of basic concepts like ‘equals’, ‘the’ and so on… lots more.
…
Seriously: I don’t think things are going to be a big complicated mess when we sort them out. We just need to expect that our future ways of talking will be quite different than the way we talk now. Have you ever read, oh, say, Diophantus? His stuff is really hard to read.
Thanks for these comments. Feeling more inspired now. It is good that one can obtain “categorotherapy” from the n-café!
The example of versus is indeed a great one.
Bruce once mentioned that to me in an email. But now I realize I forgot the details again.
Could anyone explain the details of that again?
And: is the subtlety really the associator in , or is it something also sometimes referred to as an associator, but really describing the relation of choices of bases for Hom-spaces of three objects in terms of Hom-spaces of two objects?
By the way — the big question about ‘principle bundles’ and their ‘moral fiber’ is whether Serre saw the exchange Allen had with Bruce here, or whether he saw that joke somewhere else.
According to the famous Google, the top-ranked reference to this joke seems to involve someone who stole it from Allen!
A few months ago, after (unsuccessfully) wrestling with the proper formulation of a type theory with a propositional equality, I mentioned to a professor of mine that I no longer understood the concept of equality. He chuckled, and said that now at last I was a genuine type theorist. The fact that real mathematicians start seeing the same sorts of difficulties arise is actually reassuring, because it’s evidence it’s a real problem and not just us type theorists making life hard for ourselves.
In fact, I think there’s a lot more to be said even in the case of the simply typed lambda calculus and bicartesian closed categories; categorists habitually define arrows modulo beta-eta equivalence, but actually formulating that in an algorithmic way reveals a lot of structure and beauty that a brutal quotienting obliterates.
Neel wrote in part:
[C]ategorists habitually define arrows modulo beta-eta equivalence, but actually formulating that in an algorithmic way reveals a lot of structure and beauty that a brutal quotienting obliterates.
This is something that we understand here at the n-Category Café … emphasis on the n!
Indeed, that is much of the point of John’s course on Classical versus Quantum Computation.
I just started to look at those – that’s some really nice stuff. Thanks!
As I was starting out with homological algebra, the lecturer telling me about it introduced a
Meta-theorem of diagram chasing: If you can trace your way through arrows from one point in a commutative diagram to another, then most probably, this particular path gives you a well-defined map from where you started to where you ended.
It was never claimed to be in any way robust, however, it helped adjusting yourself to the mindset hearing it spelled out at least once.
I just watched through the video again and caught something I can’t believe I missed before. Like Bruce above, I’m really upset at Serre’s treatment of “identifications”, but on more philosophical (specifically structuralist) grounds.
Serre mentions questions like “what is the cardinality of 3?” He gets the right answer, but immediately backs off from it. He says that the different answers “mean you should not ask the question” but that “it is not quite compatible with what we say to tell other people ‘there are questions you should not ask’”.
The problem is that this is the Caesar problem all over again, which (at least in my view) has been asked and answered. Within none of the number systems Serre mentions is “cardinality” a property of a number, so we really can’t sensibly ask what the cardinality of 3 is. Even within Zermelo-Fraenkel set theory there are all sorts of different models of the Peano axioms which assign different cardinalities to their representatives of 3.
Serre drives the nail true with “we should not ask that question”, but doesn’t stop to see that he has and proceeds to hammer down the wall.
Yes, I too found Serre’s remarks on the inclusions
to be off the mark. He seems to have found the problem with the old-fashioned set-theoretic foundations, but not the solution. He’s not taking a sufficiently structuralist approach to mathematics.
We should say that is a subset of . But of course, the mathematically useful definition of ‘subset’ is ‘subobject in the category ’. And we shouldn’t care that Bourbaki’s definition of gives a set that’s not equal to someone else’s — it’s good enough that they’re canonically isomorphic. The question ‘what is the cardinality of the number 3?’, not being invariant under these canonical isomorphisms, is not one we should be interested in.
The point I took from this part of his talk was that you should generally give morphisms names, not leave them implicit. (By the way, since we’re on such topics, I’ve never known how to punctuate that kind of construction. The only option besides a comma that seems reasonable to me is an M-dash.) Thus it is not best to say “N a subset/subobject of Z”. Indeed in the category of monoids, there are many useful ways of making it a subobject. I think he also said that it is absurd to apply this standard without compromise. But you should aspire to it and only cut corners where you have a good reason to. I now think I’ve been using constructions like “the map X–>Y” too much. I think all the business about What is the cardinality of 3? was just an old-fashioned way of making this point. He even says that this is a question we all agree you’re not allowed to ask.
In fact, not only did I find the point about not making identifications useful, for me it was the only useful one he made. An hour-long video of one of the best mathematicians and expositors of the 20th century, and we get little more than schoolbook advice on English usage!
He even says that this is a question we all agree you’re not allowed to ask.
Yes, and then goes on to say there’s something wrong with saying that. He doesn’t turn around and ask why we shouldn’t ask the question, which would have led philosophically into structuralism and mathematically into categorification. He just said that it seems inconsistent to even say there are such questions.
An hour-long video of one of the best mathematicians and expositors of the 20th century, and we get little more than schoolbook advice on English usage!
I’m not sure what the event was, but I noticed that the audience as a whole seemed to be extremely young. He said at the beginning that he was asked to talk about how to write mathematics, and I’m getting the impression that the audience was predominantly graduate students.
He was sharing his lifetime of experience as a writer with those just starting out their professional lives – a practice I wish was far more common than it is. The simple observation is that young mathematicians are not trained to write, but merely to pick up writing on the side. Further, their general educations have been slipping further and further behind in general English usage.
A paper casually written can easily fall into a number of particular logical traps in mathematics. He chose to highlight common mistakes with a particular emphasis on what can go wrong when they occur in a mathematical paper. I’m sure, James, that your English usage skills are completely above reproach and that you have absolutely nothing to add to your perfect grammar, but you are far from the average in that respect.
Aaa! I definitely didn’t mean to come across that way, though looking back at what I wrote, I can see how I might have. (Probably the M-dash comment didn’t help…) It’s always hard to predict how what you write is going to sound to others, I guess.
Now that I’ve thought about it a bit more, I do think he said lots of good things, but I suppose that since I had heard most of them long ago (from my thesis advisor, in great red lengths on drafts of my thesis), I forgot that you still have to be told a first time.
But still I think there is something odd about Serre lecturing the department at Harvard on the importance of not confusing it’s and its. Surely you can grant me that, no? I mean, imagine Milnor lecturing at IHES on the importance of not confusing le and la. Emoticon.
Well again, I’m not aware of the provenance of the lecture. You say it’s at Harvard, and I’ll take your word for that. Still, the audience looks very young, so I’m wondering if it was aimed specifically at graduate students or the like.
The video is from a talk in the Harvard “Basic Notions” seminar; there was a talk “Writing mathematics?” by Serre in this on November 10, 2003, according to the index at the Harvard math department web site. I can no longer find the video linked from their web site, and also the link to the “Basic Notions” seminar off the seminar page no longer works. (There were formerly a bunch of quite nice expositions of various topics here.) I believe the series to have been aimed at beginning graduate students.
But still I think there is something odd about Serre lecturing the department at Harvard on the importance of not confusing it’s and its. Surely you can grant me that, no? I mean, imagine Milnor lecturing at IHES on the importance of not confusing le and la. Emoticon.
However, le and la are normally confused by non-francophones - just as with der/die/das in German - and then does not change the semantics significantly. They occupy the very same language slot.
As for it’s vs its, these are semantically highly different words, and very commonly confused even among native english speakers.
Given that, I’d say that the comparison is flawed.
I’m glad someone took James to task, but then others picked up on the it’s vs its issue, which was hardly his main point!
I’ve recommended elsewhere that grad students before they start writing up their thesis results be exposed to this video.
jim
James wrote:
Now that I’ve thought about it a bit more, I do think he said lots of good things, but I suppose that since I had heard most of them long ago (from my thesis advisor, in great red lengths on drafts of my thesis), I forgot that you still have to be told a first time.
Ah, there’s the rub! As editor and referee of thesis derived papers, it is painfully obvious to me that many students do not have the benefit of ‘great red lengths on drafts of my thesis’. Frank Peterson claimed that Steenrod put him through half a dozen drafts.
While we are on the subject of ‘impoverished’ grad students, I am often solicited for letters by neophyte mathematicians who claim next to their thesis advisor, I’m the only one who has responded in detail to what they write. Not to brag, but to encourage others to share the burden.
jim
John said:
The problem is that this is the Caesar problem all over again, which (at least in my view) has been asked and answered.
John, what’s the Caesar problem?
The Caesar problem was a question raised by Frege. I believe it was an attempt to show that strict Platonism leads to seeming nonsense.
Under a strictly Platonic-realist mathematical ontology, mathematical objects actually exist in the same sense as horses and rocks and Julius Caesar actually exist. Somewhere there is “the number 3” as an actually existing thing. So it’s a well-posed question to ask “is 3 equal to Julius Caesar?”
What I’m saying is that when you understand that there is no one “3” – that “3” is a slot in the natural number structure defined by the Peano axioms rather than a member of any particular model of the structure – then you see that the only questions that can be sensibly asked about 3 are those defined by the natural number structure. Cardinality and identity with ancient Roman emperors are not among those questions.
You may as well ask “Is the calculus professor a woman?” This professor or that one may be a woman, but sex is not a property of the role of calculus professor, just of the role-filler.
That’s really not what the Caesar problem was about. Actually it concerns Frege’s claim that the definition of an entity such as a number should be given in such a way that we can tell that Julius Caesar is not a number. It is precisely because we should be able to ask questions such as ‘Is Caesar the same as the number 3?’, and that we should be able to answer ‘No’ from the definition of 3, that we had better define ‘3’ the right way.
See this entry in the online Stanford Encyclopedia of Philosophy.
On one hand, this is an issue of whether a language is or is not strongly typed. That relates to “category errors” in Philosophy, which may or may not have anything at all to do with Category Theory.
On the other hand, fo at least some subset of people, numbers ARE in the same category as people. In particular, to cite one example, the functioing autistic synaesthetic or whatever author of the autobiography “I was born on a blue day” explains at length how numbers were his close friends, always the same, reliable, having personality. My wife, a Physics professor, commented when this author was interviewed on TV, “That’s interesting, I was born on a Green Day.”
Anecdotally, Ramanujan also in a sense “knew” integers under 10,000 as individuals with personality, hence the Taxicab Numbers story. There’s some evidence that Gauss also “saw” integers since, as a child, he made figurate numbers with pebbles while at the site of the construction projects his father supervised.
Vladimir Nabokov, a scientist (though not a Mathematiciasn) also saw each number as having distinct sensory characteristics, and each letter. he particularly admired ONLY the scene with the hookah-smoking caterpiller in the Disney version of “Alice in Wonderland” for how his spoken words had letters which became smoke-ring shapes that interacted with things and people.
So it not trivial to say that Julius Caser and 3 cannot be the same. Depends on your model, and you metamodel that includes models of people whose brains are wired very differently.
And there is a body of Mathematical Science Fiction which considers other axioms, other connections between axiomatic and empirical truth, and other intelligences who apprehend numbers very differently than humans do.
But this is not the venue for me to argue this in depth, only to say that the Caeser problem is nontrivial.
Jonathan wrote at last:
And there is a body of Mathematical Science Fiction which considers other axioms, other connections between axiomatic and empirical truth, and other intelligences who apprehend numbers very differently than humans do.
I’d be interested in recommendations for such stories, especially (but not only) the last. (I strongly doubt the assumption that technologically advanced non-human intelligences must apprehend the ‘natural’ numbers much like us, but I need to expand my mind with the possibilities.) Feel free to email me (through my web page if you know no better way) any replies if you consider them too far off topic.
I don’t know about the Caesar problem, but it sounds like John Armstrong and David Corfield are not disagreeing as much as they sound like they are.
A nice solution of the Caesar problem is to use a typed system of logic, where each entity has a ‘type’. In many typed systems it’s simply forbidden to ask if two entities of different types — like a person and a natural number — are equal. So, the answer to ‘is Caesar equal to the number 3’ is not just ‘no’, it’s more like ‘Bzzzt!’
Toby Bartels is the local expert on type theory, but I’m teaching a course on the typed lambda-calculus just now, so this stuff is on my mind.
In the category-theoretic approach to logic, types correspond to objects and ‘entities of type ’ correspond to morphisms . We can formalize category theory in such a way that ‘equality’ only applies to parallel arrows , not to non-parallel arrows or (heaven forfend!) objects. So, one just can’t ask if ‘Caesar equals 3’ or if ‘the type People equals the type Natural Numbers’.
It seems that Serre understands perfectly well that one shouldn’t ask certain stupid questions — but he also sort of snickered at this idea, in a way that suggests he’s unaware of foundations of mathematics that formalize this idea.
I don’t know about the Caesar problem, but it sounds like John Armstrong and David Corfield are not disagreeing as much as they sound like they are.
I was disagreeing with John Armstrong about what he understood Frege meant by raising the Julius Caesar problem. I hope it doesn’t bother him me saying so, but he couldn’t have been more wrong.
A very different issue is how we might want to respond to what Frege sees as a problem, and there I dare say we would find ourselves in close agreement.
You, John B, raise the issue of types, and I agree that is the way to go. As I pointed out in this post, many philosophers want nothing to do with types, but would rather have a set of all ‘possibilia’ to quantify over.
I’m a computer scientist (although not logician), I’m not convinced that for practical construction of representations of knowledge that typedness really works, the worry coming from granularity and the notion that you can specify the types of an object as soon as you have an instance of it. (Clearly multiple-typedness occurs in general as, e.g., I’m as alive as earthworm X (so we must share a type for that relation to be well-defined) but “Am I more compassionate than earthworm X” doesn’t seem defined, so we must be of different types for “compassionateness” comparisons.) But there are lots of questions which “to human minds” make sense that seem to imply incredible granularity of typedness, eg, “am I fatter than that” makes sense for the collection of particles making up a statue of Churchill (which might be a different answer from “am I fatter than Churchill” depending on quality of statue) or the collection paint particles depicting Nelson. So you’d be tempted to define a type “person or person representation” for the “fatness questions” suitability. But what about the mother-in-law invoked by comedians with a line “my mother-in-law is so fat that Y”? Well, given that Y doesn’t apply to me it’s a reasonable conclusion that “I’m not fatter than the mother-in-law”. But now the type has expanded to “person or person representation or imaginary construct defined to have an attribute relevant to fatness”. And “am I fatter than James Bond”, who’s “a fictional character with multiple representations”, all of whom I am fatter than, so we expand the type… etc ad nauseam. And I’m certain no-one has thought of the “comedian’s mother-in-law in joke X” as an instance of a “person or person representation or imaginary construct defined to have an attribute relevant to fatness” until they read this post even though they’ve probably heard one such joke, so you’ve had an instance of the type before you had the type itself.
I concede it’s quite likely possible theoretically have infinitely fine granularity types and infinitely precise query specifications so that typedness does weed out “ill-formed” questions, but from the point of view of creating artificial intelligence it seems some more “on-the-fly constructional technique” for what makes a sensible logic seems more applicable, and I suspect such a thing ought to be of at least partial interest to philosophers/logicians.
At least, that’s my uninformed view.
Dave tweed wrote a comment largely summed up by:
But now the type has expanded to “person or person representation or imaginary construct defined to have an attribute relevant to fatness”.
I think that it’s a mistake to try to define a type of all objects whose fatness may be sensibly compared to a human’s. Rather, you should define a very abstract type of humanoid fatnesses. Then various other types (the type of humans, the type of statues of humans, the type of fictional human characters, etc) may be equipped with a fatness map to this abstract type. (This is, I think, Lawvere’s view, that a property of some type is a map from a concrete type to an abstract type of possible values of the property.) As you expand your range of objects whose fatness you want to compare to a human’s, you add further maps from appropriate types to the abstract type of fatnesses. Of course, at any stage you could form the sum of all types equipped with such fatness maps, to get a temporary type of objects with human fatnesses, but this is not really necessary. (Incidentally, I use the word ‘humanoid’ to describe this type only because the first fatness map is from the type of humans. If you started with, say, incarnations of James Bond, you might call it something else.)
I’m sure that there’s better ways of ensuring typedness. I was trying to make more of a meta-comment along the lines of “when you’re figuring out how to type things you’re solving the conceptually difficult problem, then the actual types you’ve ended up with streamline actual calculations/reasoning”. Certainly in general discourse (ie, not just mathematics) you sometimes ask and answer questions for which it seems unlikely you have previously defined types. In particular, I was idly wondering if there’s any deep, primarily mathematical aspects to the “figuring out types in the first place” process. E.g., the question of whether 1+i is less than 2+3i is “ill-defined because complex numbers aren’t an ordered field”, but the “most interesting bit” is how you decide things are ordered or cannot be ordered, and maybe there’s something mathematically deep about that process. Or maybe there isn’t. Or maybe we’re centuries away from formulating it.
Dave tweed summarised his own comment:
“when you’re figuring out how to type things you’re solving the conceptually difficult problem, then the actual types you’ve ended up with streamline actual calculations/reasoning”
I agree with that! Once you’ve got things properly typed, then both proofs (in mathematics) and programs (in computer science) are often simply a matter of ‘following your nose’, as John puts it.
I may have the original motivations muddled, but I think John is right that we aren’t really disagreeing here. To recast what you’re saying, Frege thinks that it’s necessary to have the right definition of just what “3” is because the question “does 3 = Julius Caesar?” is a valid one that needs to be answerable.
JVP and John have both brought out type theory as a way to make the answer immediately come out “no”, but I don’t even think you need that much machinery. Actually, I don’t think the answer should be “no”. It should be MU!. The question is ill-posed from the beginning, and to even try to make the answer come out as “no” (say, with type theory) says that the question is well-posed.
I’m sure one could construct some particularly weird model of the Peano axioms where the role of 3 was filled by Julius Caesar. I definitely can construct a model where the role of 3 is filled by a set with arbitrary cardinality. Cardinality and being Julius Caesar are simply not properties the Peano axioms discuss, and so they simply don’t apply to the natural numbers.
The reason that Serre’s answer seems plausible is that we tend to think in set theory, and have this idea in the backs of our minds that everything is “really” a set, and so has cardinality. But all we really should think is “everything (or at least most things) can be modelled inside of set theory”. There are all sorts of other properties we could ask about that should all come back with the answer, “don’t be silly.” What is the dimension of 3? What is the characteristic of 3? What is the fundamental group of 3? Since we don’t think of 3 as a vector space, a field, or a topological space, it’s clear that these questions just don’t apply.
John Armstrong wrote:
JVP and John have both brought out type theory as a way to make the answer immediately come out “no”, but I don’t even think you need that much machinery. Actually, I don’t think the answer should be “no”. It should be MU!.
In case anyone is wondering, ‘mu’ is a Zen way of saying ‘unask that question — it’s based on incorrect assumptions!’
And, type theory actually lets the answer come out ‘mu’, not ‘no’. In a strongly typed logic, it’s simply impossible to assert or deny the equality of entities of different types! That’s what I meant when I wrote:
So, the answer to ‘is Caesar equal to the number 3’ is not just ‘no’, it’s more like ‘Bzzzt!’
However, ‘mu’ is more precise than ‘Bzzzt!’
Dave Tweed is of course correct that there are serious problems with typedness in the “practical constructions of representations of knowledge”: it seems like there are lots of types of things, but their borderlines are fuzzy and ever-changing — and as we keep learning stuff, new types get born and old ones die! So, a strict system of types may not be worth the hassle, except in limited realms of discourse.
Lewis Carroll’s White Knight pointed this out quite nicely:
He thought he saw a Rattlesnake
That questioned him in Greek:
He looked again, and found it was
The Middle of Next Week.
and later:
He thought he saw a Argument
That proved he was the Pope:
He looked again, and found it was
A Bar of Mottled Soap.
In math things are a bit less unruly — at least within a fixed formal system.
I don’t recall the origin of a fallacy or logic puzzle which involves, I think, problems of isomorphisms and equality.
It goes something like this (please correct me if you know the proper citation and example):
“The first President of the United States” = “George Washington”
Law of substitution: if two objects are equal, once can be substituted for another.
“George Washington” has 16 letters.
Therefore, “The first President of the United States” has 16 letters.
I know it’s an old fallacy, and based on confusion of the name of an object with the object. But isn’t it an example of the problem cited about “up to isomorphism” and the like?
Jonathan wrote:
I know it’s an old fallacy, and based on confusion of the name of an object with the object. But isn’t it an example of the problem cited about “up to isomorphism” and the like?
No! It’s just an example of mixing up something with its name. Again, Lewis Carroll’s White Knight gives the definitive treatment:
‘You are sad,’ the Knight said in an anxious tone: ‘let me sing you a song to comfort you.’
‘Is it very long?’ Alice asked, for she had heard a good deal of poetry that day.
‘It’s long,’ said the Knight, ‘but very, VERY beautiful. Everybody that hears me sing it–either it brings the TEARS into their eyes, or else–’
‘Or else what?’ said Alice, for the Knight had made a sudden pause.
‘Or else it doesn’t, you know. The name of the song is called “HADDOCKS’ EYES.”’
‘Oh, that’s the name of the song, is it?’ Alice said, trying to feel interested.
‘No, you don’t understand,’ the Knight said, looking a little vexed. ‘That’s what the name is CALLED. The name really IS “THE AGED AGED MAN.”’
‘Then I ought to have said “That’s what the SONG is called”?’ Alice corrected herself.
‘No, you oughtn’t: that’s quite another thing! The SONG is called “WAYS AND MEANS”: but that’s only what it’s CALLED, you know!’
‘Well, what IS the song, then?’ said Alice, who was by this time completely bewildered.
‘I was coming to that,’ the Knight said. ‘The song really IS “A-SITTING ON A GATE”: and the tune’s my own invention.’
Interesting, but I guess not surprising, that almost all the comments have focused on the identification and equality issues.
Notice the admission that Bott and Tu is comprehensible as opposed to the Chevalley exercise in rectitude of thought.
What I focused on instead - as editor and referee of too many years was (as I wrote to Serre):
Cher Serre,
Je viens de decouvrire votre video “How to write mathematics badly”. Formidable! As editor/referee/mentor, I have long had some of the pet peeves that you mention. A few of yours struck a particularly strong chord.Terms with multiple definitions. My bete noir is ‘twist’ or ‘twisting’; I once tried to invent a neologism for Drinfel’d’s twisting: skrooching but it never caught on.
Milnor taught me: Never start a sentence with a symbol.
Moise referred to ‘proofs of infinite thinness’; in which the hole was as big as the proof.
Ah, the problem of signs; what grief they have caused. I forget who phrased the metatheorem: There exists a set of signs.
I am delighted with your definition of principle bundle.
Pyranian took exception to abbreviations of Latin terms: e.g. mut. mut., assuming rightly that most of the younger generation had no knowledge of Latin.
Commas: I am infamous for insisting on correct English use of commas, as opposed to sprinkling them at random. But then both my parents were English teachers.
References: Many physicists do not list titles in their bibliography.
At one conference, the organizers insisted that papers submitted for the proceedings include titles. One participant responded: what if you don’t know the title of the paper!!
I suggest every grad student just before the writing part of his thesis watch this video.
jim
Mostly because I actually enjoy nitpicking to a certain extent: I find fault with the statement that the younger generation - in its entirety - is unfamiliar with Latin. Indeed - neither Serre, nor you, Stasheff, seem to be quite able to spell out mutatis mutandis reliably.
There are, still, reasons to object to a lax use of mutatis mutandis - but lack of knowledge on behalf of my generation at least I emphatically deny.
As for commas, I’m now in my third language tradition, and the third different set of rules for commas. I’m lost when it comes to keeping them quite straight. Alas.
When Pyranian implicitly accused the younger (than his) generation of ignorance of Latin, of course he didn’t mean all and he probably had Americans in mind more than well educated Europeans. Actually, he was objecting to the use of abbreviations of Latin - hence my writing mut. mut. Another he objected to was N. B. for Note Bene. I also diagreed with Serre’s transltion of mutatis mutandis.
My sympathies on the comma issue for those for whom English ‘was not the language the muse sang at their cradle’. Even more sympathy for those who have trouble with a, the and \null - primarily Slavs but not exclusively. And for all those whose native language has only one word for recall and remind.
jim
Another he objected to was N. B. for Note Bene.
Nota bene, isn’t it?
Let us show mercy to our fellow pedants, Todd.
Surely we can all agree with Serre’s general point about the dangers of conflating “a” with “the”. This ties in, in a way, with this business about making identifications, although I’m not sure Serre quite made the connection explicit; I’d have to check. It’s where one makes a blithe identification when the structures involved do not determine a “canonical” choice of isomorphism (bandying about that much-abused word “canonical”; mostly I mean something determined by a universal property). But I generally don’t mind such identifications when there is a canonical isomorphism.
Thus, I would say that Serre’s example of the subset identification as being abusive is somewhat superficial. I would think that writing the subset symbol in the situation where we are considering and as rigs is completely harmless, since there is only one choice of injection. The point about there being no literal inclusion according to some particular encoding or other of these structures in ZF is hackneyed and not really to the point.
Thus, , , and have an internal rigidity (no nontrivial automorphisms) which makes it entirely permissible, in my view, to speak without danger of the rationals, the reals, etc. (again, as rigs).
But there are lots of situations where identifications of the abusive kind are made. How often have you seen talk of “the” field with elements? Any two are isomorphic, of course, but how – which isomorphism? A similar example that Jim Dolan recently complained to me about: some author had invoked an identification of the algebraic closure of the field of -adics with . Can you even construct one without AC, much less without having to choose one?
Elsewhere in his talk, Serre mentioned a pet peeve of mine: mixing up “it’s” with “its”. Even or perhaps especially when it comes from someone I respect greatly (including someone here at the Café, not to name names), my estimation winds up being lowered somewhat. Please don’t make that mistake!
It’s probably a good idea not to confuse typos with mistakes. Speaking for myself, I tend to make homophone replacements when typing. They can get much more bizarre than a simple they’re/their replacement or some such. I know other people who do so, too.
Yes. I’m prone to that sort of thing too. I primarily have in mind people who make this mistake consistently.
Then again, in this venue one generally has a good chance of avoiding errors of this sort just by previewing and reading what one wrote.
Aaron spoke for himself:
I tend to make homophone replacements when typing.
Me three! (Does this post get cancelled by the moderator?)
Since I read primarily by forming sounds in my head[1], this is even easy to miss when proofreading.
[1] This includes foreign (that is non-English) languages, which I will sound out before attempting to understand (just as in English, really, except that English is faster). For reading mathematics, this is tricker, since many symbols have multiple pronunciations to go with their multiple meanings; sometimes I have to backtrack and read it again. For reading Chinese (which I do very badly if at all), I tend to sound characters out in English, since I remember the pronunciation (in Mandarin) of only about half of the characters whose meaning I remember (and I remember the pronunciation of only characters whose meaning I remember, so Chinese is opposite in this respect from all the other foreign languages that I can partially read).
Since I read primarily by forming sounds in my head[1], this is even easy to miss when proofreading.
Interesting! I certainly subvocalize as I write, since I’m trying to capture to some degree the intonation I might use when speaking aloud, but the sounds in my head are far less pronounced when I read.
I hesitate to harp on this it’s/its thing any more than I have already; it’s not my goal to earn a reputation as some sort of language cop. I’ll just suggest, for anyone who has trouble with it’s/its: say to yourself ‘it is’ (or ‘it has’) whenever you see ‘it’s’. Hopefully, after a few repetitions, it will stick.
(Note to Terry: your explanation may be the right one in a lot of cases, but I’ve also seen ‘its’ used in place of ‘it is’.)
A more amusing example of homonym replacement I’ve seen:
I could of told you that that was wrong!
Here it seems to me to be a case of being almost completely blind to what one is writing, no?
Bah. Homophone, not homonym.
I’m turning in my badge.
>Elsewhere in his talk, Serre mentioned
>a pet peeve of mine: mixing up “it’s”
>with “its”.
Not to defend bad grammar, but this
particular error is more forgivable
than most, because it follows from
naive extrapolation of valid grammar.
Starting from correct examples such
as
An object that belongs to John = John’s object.
An object that belongs to a person = a person’s object
An object that belongs to the people = the people’s object
one might expect
An object that belongs to “it” = it’s object.
Of course by considering what goes on
with other pronouns, one can see that
the rule doesn’t extend from nouns
to pronouns; we use “my” instead
of “I’s”, etc. Nevertheless, the
rule “the genitive suffix is ‘s” holds
in 90% of cases in English, and it is
therefore not surprising that it is
sometimes misapplied to the remaining 10%.
Actually I found Serre’s talk very well
balanced. A good talk needs to collect
both trivial examples and deep examples
of the subject of discussion for maximum
pedagogical effect.
David Goss has some advice about writing mathematics, which apparently owes something to Serre.
Where?
jim
Just click on where I wrote ‘advice’. Or in full,
http://www.math.ohio-state.edu/~goss/hint.pdf
The video link in the original post has rotted with age, but the talk appears to be viewable here.
Thanks! I updated the link!
Re: How to Write Mathematics Badly
I haven’t found the hour to watch Serre yet, but I still want to mention Mathematics Made Difficult, which I’ve never gotten to look at.